Energy Conservation Exercise with Spring and Spider

What are the energy values involved in a spring and spider scenario?

a) How much energy is stored in the compressed spring?

b) How much potential energy did the spider initially have while sitting on the spring?

c) What was the initial kinetic energy of the spider before the spring is released?

d) How high above the ground does the spider shoot before falling back to earth?

BONUS: What is the maximum speed of the spider during this process?

Answer:

a)  [tex]k_{e}[/tex] = 928 J b) U = -62.7 J c) K = 0 d) Y = 11.0367 m e)  v = 15.23 m / s

Explanation:

Energy Conservation Exercise

This problem required the understanding of energy conservation. When a spring is compressed, it stores potential energy. After release, the stored energy is distributed into kinetic and potential energy, making the total energy conserved.

Explanation:

The questions pertain to the concept of mechanical energy conservation. The mechanical energy of a system is conserved if the only forces doing work are conservative forces. In this case, the spring force and gravity are the conservative forces.

(a) The energy stored in the compressed spring is calculated using the formula for the potential energy of a spring, which is U = 0.5 * k * x^2, where 'k' is the spring constant and 'x' is the displacement of the spring from its equilibrium position. Therefore, U = 0.5 * 2900 N/m * (0.80 m)^2 = 928 J.

(b) The initial potential energy of the spider while sitting on the spring is due to its height above the ground, which is 0.80 m, so potential energy = mass * gravity * height = 8 kg * 9.8 m/s^2 * 0.80 m = 62.72 J.

(c) The initial kinetic energy of the spider before the spring is released is 0 J because the spider is not moving at this point.

(d) The maximum height the spider reaches is found by equating the total mechanical energy at the beginning (spring potential energy + gravitational potential energy) to the total mechanical energy at the maximum height (kinetic energy + gravitational potential energy). Since the kinetic energy at the maximum height is 0 (the spider momentarily stops before falling back down), the maximum height equals (stored energy in the spring + initial potential energy) / (mass*gravity) = (928 J + 62.72 J) / (8 kg * 9.8 m/s^2) = 11.0367 m.